This achievement will be a long, long grind so here is what to expect regarding this:
100,000 pins: Each game of versus at 4.0 average power cost I was using about 25 pins per match. Note, that multiple character pins do count as defeating multiple pins (thanks to ChrisD0811 for the confirmation. Although the achievement states to defeat pins, it will roughly be about the same as what you play so for raw numbers, we will go with 25-30/match ymmv. For Horde, it is a little different because of two things: 1. what percent of the horde you kill vs your partner and 2. what type thumper you use. So far I have had 50% 2 wave thumpers and 50% 3 wave thumpers. Maybe I am not high enough level but those are the only two I have seen (Bronze League level 10). For this guide we will use a 2.5 wave average which equates to about 30-40 pins defeated per match which is about 12-20 pins per wave (we will use 20 to be optimistic) but once again ymmv. Since you have to win 1000 versus matches and clear 1000 horde waves you are looking at:
1000 x 30 = 30,000 for wins
1000 x 20 = 20,000 for cleared waves
Assuming a 80% wave clear rate and a 50% win rate:
1000 x 30 = 30,000 for losses
200 x 20 = 4,000 for uncleared waves
This totals to be about 84,000 pins so there is still a little ways to go so if by now you have not reached it just continue playing as you see fit.
Edit: Also, you will get more pins than the suggested above because wretches, snub soldiers, lancer crew, gnasher gang, etc count as multiple enemies. This will add probably close to 5-10 (we will use 8 as an average) more each match than the above suggestions so:
2000 games at 8 more pins =16,000.
Coincidentally, that 84,000 and 16,000 totals the 100,000 needed.
The next phase of this will be reaching Gold in versus. This will be a pain, because you cannot just grind this out, you actually have to be very skilled at the game. Gold league requires 2400 cogs and once you hit Bronze you will lose roughly about the same amount of cogs as if you were to win. This means you will have to win ~25 more games than you lose (starting at Bronze) to reach this league. Each season (so far) looks like they will last over a month, so there is plenty of time to reach it if you are persistent and skilled. Note that if you are in a Bronze or higher league you will not go below 1599 cogs once the season resets, so it will not be a complete loss.
Edit: Keep in mind that you will possibly have to drop serious cash to reach this league. I am having trouble getting to Silver league and I spent $15 (Luckily I had some google play money) and have hit a plateau. Also, over time the f2p people will start to cap out their pins so as long as you are persistent you will eventually be high enough ranked to perhaps reach this league. Who knows, maybe TC will adjust the cogs needed each season. We will find out end of September.
You can track this by going to Missions -> Achievements -> Seriously POP! at the very bottom.
You can track the versus matches won by going to your player profile and looking under your statistics to see how many matches you have won.
You can see if you have reached the highest league by the League Legend achievement.
Horde waves cleared can only be tracked up to 500 by the Endless Summer achievement.
As of now there is no separate way to track the defeated pins count except for the Seriously POP! achievement tracker in game.
I would appreciate comments that show what builds you are having success with. I know that depending on your level some pins will be better than others, but right now this is what I am running with their level beside them:
Dom - 4
Grinder - 3
Reyna - 4
Old Man Marcus - 1
Marcus - 3
Ink Grenade - 9
Gnasher Gang - 5 (I swap this out occasionally maybe for Kait, JD, or Longshot)
Snub Soldiers - 10 (I swap these out often for something else to try out)
EDIT Sep 12, 2019: Almost all pins have been either nerfed or buffed so this loadout is not the best at the current state of the game. I am experimenting to find out the best combination for the new pin stats.
Hopefully this guide gives you a little insight so please leave me a comment for something that would be beneficial to add to this solution. Good Luck!!!